Classical Mechanics on Noncommutative Space with Lie-algebraic Structure

Abstract

We investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results {\em not only} include that of a recent work as our special cases, {\em but also} provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable $t\dot{x}$-, $\dot{(xx)}$-, and $\ddot{(xx)}$-dependence besides with the usual $t$-, $x$-, and $\dot{x}$-dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle's ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.

21 pages, 4 figures; v2: minor clarification and two references added; v3: 22 pages, clarifications added; v4: 23 pages, clarifications added; v5: minor layout revision, final version accepted by Annals of Physics

References in corpus (10)

• Twisting all the way: from Classical Mechanics to Quantum Fields
• Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton-Hooke symmetry
• Newton equation for canonical, Lie-algebraic and quadratic deformation of classical space
• Kappa-deformed oscillators, the choice of star product and free kappa-deformed quantum fields
• Canonical and Lie-algebraic twist deformations of Galilei algebra
• Tensor Coordinates in Noncommutative Mechanics
• Canonical and Lie-algebraic twist deformations of $\kappa$-Poincare and contractions to $\kappa$-Galilei algebras
• Star Product and Invariant Integration for Lie type Noncommutative Spacetimes
• Canonical, Lie-algebraic and quadratic twist deformations of Galilei group
• Noncommutative Lagrange Mechanics

Cited by in corpus (15)

• Parameters of noncommutativity in Lie-algebraic noncommutative space
• Classical dynamics on curved Snyder space
• Classical mechanics of many particles defined on canonically deformed nonrelativistic space-time
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Closedness of orbits in a space with SU(2) Poisson structure
• Variational problem for Hamiltonian system on so(k, m) Lie-Poisson manifold and dynamics of semiclassical spin
• Non-Grassmann mechanical model of the Dirac equation
• Central force problem in space with SU(2) Poisson structure
• Generating of additional force terms in Newton equation by twist-deformed Hopf algebras and classical symmetries
• Bianchi type I cyclic cosmology from Lie-algebraically deformed phase space
• The Henon-Heiles system defined on Lie-algebraically deformed Galilei space-time
• Eigenvalue problem for radial potentials in space with SU(2) fuzziness
• Features of description of composite system's motion in twist-deformed space-time
• Twist deformations of Newtonian Schwarzschild-(Anti-)de Sitter classical system
• Twist deformation of doubly enlarged Newton-Hooke Hopf algebra